Help factoring M127^k-1

[fivemack]

Originally Posted by swellman

I’m not ashamed to admit that cyclotomics are not intuitive to me. Just to be sure I understand the naming convention, are the following correct?

C142_M31_c38

C743_M127_c50

Or should both polcyclo indices be half the values listed (i.e.19, 25)?

(Plus I get wrapped up sometimes with factordb not always recognizing composite factors previously discovered elsewhere - up or down - in a poly chain. No doubt this skill comes with experience.)

Those names are correct. You could check (and indeed I have just checked) by using gp:

polcyclo(38,2^31-1)%C142 should give 0
polcyclo(19,2^31-1)%C142 won't

[swellman]

Thanks for clarifying. Think I’ve got it now.

[kosta]

@Dr Sardonicus:
For us here, Arifeullians are not a thing, + is interesting only to the extent that it is a factor of some -, and as Robert Silverman reminded me just the other day, we should not call these numbers "Cunnigham" even though Cunnigham himself did consider b=2^7-1...

@swellman
two c's, one capital and one lowercase are going to confuse people. To illustrate what I was trying to say, this cofactor, lets call it C142_M31_k38, first appears in k=38, then reappears at k=76 (thats what fivemack was checking, the C142 is not a factor of the polcyclo(19) or of k=19). So, looking at the tables is enough to know that it is coming from the 38th cyclotomic polynomial, but its nice to check with pari/gp. So, I agree with fivemack, but let's use letter "k" instead of "c", haha ;-)

Note added:
cyclotomic comes from greek "kyklos" which is circle and "tome" which a cut or division, so letter k is appropriate.

[LaurV]

Wow! It looks like you found a mersenne prime which is also the exponent of another mersenne prime...

[swellman]

C156_M31_k25 is factored by SNFS. Reported to factordb.

p60=656850278015660339603725017552344842197915229736126332594501
p96=190674419612794339208450258722749530575116801131660918678504876289052717128834680829986573454701

[fivemack]

OK, I went to SNFS much too early. C187_M31_k23 is done

p22 factor: 2476363315760296607893
p33 factor: 525435349751786867693675066811697
p42 factor: 981837273061848593364978172320737104860391
p91 factor: 3440781771197002299203508060084665118254362958353613892296460316185452235397587206400281609

[swellman]

If anyone is interested, there’s always C177_M31_k44 available for G/SNFS. Yoyo@Home seems to have mined out all the easy factors for M31 long ago.

I’m finishing up C142_M31_c38 now.

@kosta - Are there other Mersenne bases to be considered other than M31, M61 and M127?

[kosta]

@swellman
M7 is of some interest, but this one has been done to very high powers.
I am trying yet other strategies to obtain full factorizations for M127:
running ecm on k>100 hoping to get a lucky hit on a big prime cofactor.
There is less than a one in a thousand chance of success :-(

I've also written a libpari C program to search for primes of the same form, with all prime k.
Sadly, no luck here either, all the way up to k <= 10,000

[fivemack]

Originally Posted by swellman

If anyone is interested, there’s always C177_M31_k44 available for G/SNFS. Yoyo@Home seems to have mined out all the easy factors for M31 long ago.

I'm confused as to why you say that, when I've just found some annoyingly trivial factors for M31_k23 ... I have set a fair amount of ECM going on M31_k29, that and M31_k31 would be heavy SNFS jobs but just about practical for nfs@home.

[fivemack]

I found a P30 ( 281801749001573550949215731459 ) in M31_k29 within three minutes of starting to look; it looks as if yoyo@home only looked at k not coprime to 240.

[swellman]

Originally Posted by fivemack

I'm confused as to why you say that, when I've just found some annoyingly trivial factors for M31_k23 ... I have set a fair amount of ECM going on M31_k29, that and M31_k31 would be heavy SNFS jobs but just about practical for nfs@home.

When I went poking around fdb looking at M31 and M61 it seemed as if there were many p5X and p6X factors, plus many larger composites. Looked like rubble left over after Yoyo rolled through, but from your findings those are most likely leftovers from other focused efforts rather than a wide swath of ECM. I assumed facts not in evidence, sorry ‘bout that.

C177_M31_k44 Is still an interesting target, though some ECM to t55 might be prudent!